DEBYE-HUCKEL-ONSAGER TREATMENT FOR AQUEOUS SOLUTION AND ITS LIMITATION

It is a well-known fact that the conductance of weak electrolytic solutions increases with the
increase in dilution. This can be easily explained on the basis of Arrhenius's theory of
electrolytic dissociation which says that the magnitude of dissociated electrolyte, and hence the
number of charge carriers, increases with the increase in dilution. However, the problem arises
when the strong or true electrolytes show the same trend but at a much lower scale. We used
the word “problem” because even at the higher concentration, the electrolyte dissociates
completely inferring that there is no possibility of further dissociation with dilution. This means
that there should be no increase in the conductance of strong electrolytes with the addition of
water.

Figure 14. The typical variation of molar conductance (Λm) with the square root of the
concentration (√𝑐)
for strong and weak electrolytes.

The primary reason behind this weird behavior of strong electrolyte is that the
conductance of any electrolytic solution depends not only upon the number of charge
carriers but also upon the speed of these charge carriers. Therefore, if the dilution does
not affect the number of charge carriers in strong electrolytes, it must be affecting the
speed of ions to change its conductance. The main factor that is responsible for
governing the ionic mobility is ion-ion interactions. Now since these ion-ion
interactions are dependent upon the interionic distances, they eventually vary with the
population density of charge carriers. Higher population density means smaller
interionic distances

and therefore stronger ion-ion interactions. On the other hand, the lesser
population density of ions would result in larger interionic separations and hence
weaker ion-ion interactions.

In the case of weak electrolytes, the degree of dissociation is very small at high
concentrations yielding a very low population density of charge carriers. This would
result in almost zero ion-ion interactions
at high concentrations. Now although the degree of dissociation increases with dilution
which in turn also increases the total number of charge carriers, the population density
remains almost unchanged since extra water has been added for these extra ions. Thus,
we can conclude that there are no ion-ion interactions in weak electrolytes neither at
high nor at the low concentration; and hence the rise in conductance with dilution
almost a function dissociation only.
In the case of strong electrolytes, the degree of dissociation is a hundred
percent even at high concentrations yielding a very high population density of charge
carriers. This would result in very strong ion- ion interactions at high concentrations,
hindering the speed of various charge carriers. Now when more and more solvent is
added, the total number of charge carriers remains the same but the population
density decreases continuously creating large interionic separations. This would result
in a decrease in ion-ion interaction with increasing dilution, and therefore, the charge
carriers would be freer to move in the solution. Thus, we can conclude that though
there is no rise in the number of charge carriers with dilution, the declining magnitude
of ion-ion interaction creates faster ions and larger conductance.
 1
𝐼 = (𝑚+ 𝑧+
2
1
∴ 𝐼 = (0.002 × 12 + 0.001 × 22 ) = 0.003
2

The mean activity co-efficient (γ±) is given by the equation:

𝑧+ 𝑧−

DEBYE-HÜCKEL-ONSAGER THEORY OF ELECTROLYTES

DHO theory assumes that strong electrolytes are completely ionized at all dilutions. It is
observed that, there are three types of opposing forces which oppose the velocity (or mobility)
of ions in solution in turn decreases its conductivity.
They are:

1) Relaxation Force (Asymmetric Effect)

2) Electrophoretic Force
3) Friction Force

Asymmetry Effect (Relaxation Effect)

On account of electrostatic attraction, a central ion in solution is surrounded by opposite charge

ions to form an ionic atmosphere, and it will have spherical (central) symmetry. But when an
electric potential is applied, the central ion tends to move towards the oppositely charged
electrode while the ionic cloud moves in the opposite direction of central ion. This effect creates
an asymmetry in the ionic atmosphere (Fig. 1.2).

Because of asymmetric effect, the velocity (mobility) of central ion decreases in turn the
conductance of the solution decreases. This effect is more in concentrated solution than in
dilute solution. Note that the ionic atmosphere is repeatedly being destroyed and formed again.
But the new ionic atmosphere is not formed at the same rate at which the old one disappears
and the later takes more time to form is called relaxation time. So, the asymmetric effect is also
known as relaxation effect and the opposing force for the decrease of conductance of the central
ion is known as relaxation force.

7
 Anode (+) Cathode ( )

+ +

Symmetrical Ionic Asymmetrical Ionic

Atmosphere Atmosphere

Anode (+)
Cathode ( )

+ + +

Old Ionic Asymmetrical Ionic New Ionic

Atmospher Atmosphere Atmospher

Relaxation Time
.
Fig. 1.2: Asymmetry Effect

Electrophoretic Effect

This is another force that opposes the velocity of central ion and is responsible for lowering of
its conductivity when an electric field is applied. The central positive ions move towards
cathode whereas the negative ions move in the opposite direction towards anode.

Anode (+) Cathode ( )

Fig. 1.3: Electrophoretic Effect

The ions during their movement impart momentum of solvent molecules (water) since negative
charge ions are in excess. Hence, the net momentum imparted to the solvent will have the
direction of positive electrode. That is, the streaming of central ion occurs in a direction
opposite to that of movement of solvent (water) molecules when an electric field is applied,
and the phenomenon is known as electrophoretic effect (Fig. 1.3) and the force is known as
electrophoretic force.

8
 DERIVATION OF DEBYE-HÜCKEL-ONSAGER EQUATION
Relaxation Force

Asymmetry effect is the asymmetrical distribution of the ion cloud around a central ion which
occurs from the finite relaxation time when a voltage is applied. It leads to a decrease in the
mobility of ions. Onsager showed that the value of relaxation effect (force) as:

𝜀 3 𝑍𝑖 𝜅𝜔𝑉
𝑅𝑒 𝑙 𝑎𝑥𝑎𝑡𝑖𝑜𝑛 𝐹𝑜𝑟𝑐𝑒 = ( )
6𝐷𝐾𝑇

1⁄
4𝜋𝜀 2 ∑ 𝑐𝑖 𝑧𝑖2 2
2𝑞
𝑤ℎ𝑒𝑟𝑒, 𝜅 = ( ) and 𝜔 = 𝑍+ 𝑍− and the value of 𝑞 is given by:
𝐷𝐾𝑇 1 + √𝑞

𝑍+ 𝑍− (𝜆+ + 𝜆− )
𝑞= ×
(𝑍+ + 𝑍− ) (𝑍+ 𝜆+ + 𝑍− 𝜆− )

Where, ε → Electronic charge; D → Dielectric constant of medium; T → Temperature; Ci →

Equivalent concentration of ith ion; Zi → Valency of ith ion; V → Applied potential gradient; K
→ Boltzmann constant

Electrophoretic Effect (Force)

When an electromotive force is applied on an ionic atmosphere, the central ions moves in one
direction and the oppositely charged ions present in their ionic atmosphere move in opposite
direction.

The solvent molecules attached to ionic atmosphere also move in direction opposite to that of
central ion. Thus, they cause friction due to which the mobility of the center ion is retarded.
This effect is called electrophoretic effect.

On the of Stoke’s law, Debye and Hückel calculated the following expression for the
electrophoretic force on ion of ith kind as:
1⁄
𝜀𝐾𝑖 𝑍𝑖 𝜅𝑉 4𝜋𝜀 2 ∑ 𝑐𝑖 𝑧𝑖2 2
𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑝ℎ𝑜𝑟𝑒𝑡𝑖𝑐 𝐹𝑜𝑟𝑐𝑒 = 𝑤ℎ𝑒𝑟𝑒, 𝜅 = ( )
6𝜋𝜂 𝐷𝐾𝑇

Where, ε → Electronic charge; D → Dielectric constant; T → Temperature; ci →

Concentration of ith ion in gram equivalent; Zi → Valency of ith ion; Ki → Coefficient of
frictional resistant of solvent opposing the motion of ith kind ion; η → Viscosity of the medium;
V → Applied potential gradient; K → Boltzmann constant

Frictional Force

Any moving ions experience resistive force by medium. If central ion is moving with uniform
velocity, it experiences frictional force by the solvent of the solution (medium) in turn its
velocity in the solution is decreased (or the conductivity of the solution is de creased).

9
That is, an ion with its ionic atmosphere travels in the solution, then the medium of the solution
offers the frictional resistance. This force depends upon viscosity of the medium and its
dielectric constant.
𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 = 𝑢𝑖 𝐾𝑖
ui → Steady velocity (or mobility) of ith ion: Ki → Coefficient of frictional resistant of solvent
opposing the motion of ith kind ion.

Derivation of Debye-Hückel-Onsager Equation

Debye-Hückel theory explains the determination of electrical potentials of ionic atmosphere.

In 1927, Onsager explained the various factors which are responsible for opposing the velocity
of central ion surrounded by the ionic atmosphere. The total electric force applied from external
source on ionic atmosphere is equivalent to the sum of relaxation force, electrophoretic force
and frictional force. The electromotive force applied from outside on center ion is equal to εZiV,
where ε is the electronic charge of ion of valency Zi and V is the applied potential gradient.

Electromotive force applied on center ion = Sum of all opposing forces on center ion

𝜀 3 𝑍𝑖 𝜅𝜔𝑉 𝜀𝐾𝑖 𝑍𝑖 𝜅𝑉
𝜀𝑍𝑖 𝑉 = + + 𝑢𝑖 𝐾𝑖
6𝐷𝐾𝑇 6𝜋𝜂

𝜔𝜀 3 𝑍𝑖 𝜅𝑉 𝜀𝐾𝑖 𝑍𝑖 𝜅𝑉
𝑢𝑖 𝐾𝑖 = 𝜀𝑍𝑖 𝑉 − −
6𝐷𝐾𝑇 6𝜋𝜂

𝜔𝜀 3 𝑍𝑖 𝜅𝑉 𝜀𝐾𝑖 𝑍𝑖 𝜅𝑉
𝑢𝑖 𝐾𝑖 = 𝜀𝑍𝑖 𝑉 − ( + ) ⋯ (1)
6𝐷𝐾𝑇 6𝜋𝜂

Dividing throughout the equation (1) by 𝐾𝑖 𝑉, we get:

𝑢𝑖 𝐾𝑖 𝜀𝑍𝑖 𝑉 𝜔𝜀 3 𝑍𝑖 𝜅𝑉 1 𝜀𝐾𝑖 𝑍𝑖 𝜅𝑉 1
= −( × + × )
𝐾𝑖 𝑉 𝐾𝑖 𝑉 6𝐷𝐾𝑇 𝐾𝑖 𝑉 6𝜋𝜂 𝐾𝑖 𝑉

𝑢𝑖 𝜀𝑍𝑖 𝜔𝜀 3 𝑍𝑖 𝜅 1 𝜀𝑍𝑖 𝜅
= −( × + ) ⋯ (2)
𝑉 𝐾𝑖 6𝐷𝐾𝑇 𝐾𝑖 6𝜋𝜂

Multiplying throughout the equation (2) by V, we get:

𝜀𝑍𝑖 𝑉 𝜔𝜀 3 𝑍𝑖 𝜅𝑉 1 𝜀𝑍𝑖 𝜅𝑉
𝑢𝑖 = −( × + ) ⋯ (3)
𝐾𝑖 6𝐷𝐾𝑇 𝐾𝑖 6𝜋𝜂

1
If the potential gradient is taken as one volt per cm, then 1 𝑉 = 300 electrostatic unit (esu) or
electric potential.
1
Substituting potential gradient, 𝑉 = 300 esu in equation 93), we have:

10
 𝜀𝑍𝑖 1 𝜔𝜀 3 𝑍𝑖 𝜅 1 𝜀𝑍𝑖 𝜅 1
𝑢𝑖 = × −( × + × )
𝐾𝑖 300 6𝐷𝐾𝑇𝐾𝑖 300 6𝜋𝜂 300

𝜀𝑍𝑖 𝜔𝜀 2 𝑍𝑖 𝑍𝑖 𝜀𝜅
(𝑜𝑟) 𝑢𝑖 = −( + ) ( ) ⋯ (4)
300𝐾𝑖 6𝐷𝐾𝑇𝐾𝑖 6𝜋𝜂 300
1⁄
4𝜋𝜀 2 ∑ 𝑐𝑖 𝑍𝑖2 2
Substituting the value of Debye-Hückel parameter, 𝜅 = ( ) , we get:
𝐷𝐾𝑇

1⁄
𝜀𝑍𝑖 𝜔𝜀 2 𝑍𝑖 𝑍𝑖 𝜀 4𝜋𝜀 2 ∑ 𝑐𝑖 𝑍𝑖2 2
𝑢𝑖 = −( + ) ( ) ( ) … (5)
300𝐾𝑖 6𝐷𝐾𝑇𝐾𝑖 6𝜋𝜂 300 𝐷𝐾𝑇

At infinite dilution, ∑ 𝑐𝑖 𝑍𝑖2 = 0

Substituting ∑ 𝑐𝑖 𝑍𝑖2 = 0 in equation (6), the uniform velocity (or mobility) at infinite dilution
(𝑢𝑖𝑜 ) will be:

𝜀𝑍𝑖
𝑢𝑖𝑜 = … (6)
300𝐾𝑖
𝜀𝑍
Replacing 300𝐾𝑖 by 𝑢𝑖𝑜 in equation (5), we get:
𝑖

1⁄
𝑜 𝜔𝜀 2 𝑍𝑖 𝑍𝑖 𝜀 4𝜋𝜀 2 ∑ 𝑐𝑖 𝑍𝑖2 2
𝑢𝑖 = 𝑢𝑖 − ( + ) ( ) ( ) … (7)
6𝐷𝐾𝑇𝐾𝑖 6𝜋𝜂 300 𝐷𝐾𝑇

According to Kohlrausch’s law of independent migration of ions, the equivalent conductance

of an electrolyte at infinite dilution is equal to the sum of the contributions of the equivalent
conductance of its constituent ions.

𝜆𝑜 = 𝜆+ + 𝜆−

The equivalent conductance and ionic mobility are directly proportional to each other.

𝜆+ = 𝑢+ 𝐹 𝑎𝑛𝑑 𝜆− = 𝑢− 𝐹

Equivalent conductance (λ) of any concentration is the product of mobility of ion (u), 1 mol
charge (or Faraday constant, F) and degree of dissociation (), and hence,

𝜆𝑖
𝜆 = 𝛼𝑢𝐹 (or) 𝑢𝑖 = ⋯ (8)
𝛼𝐹

The equivalent conductance (λo) at infinite dilution can also be defined for very dilute solution
as:

𝜆𝑜𝑖
𝜆𝑜𝑖 = 𝛼𝑢𝑖𝑜 𝐹 (or) 𝑢𝑖𝑜 = ⋯ (9)
𝛼𝐹

11
Substituting 𝑢𝑖 and 𝑢𝑖𝑜 in the mobilty equation (7), we get:

1⁄
𝜆𝑖 𝜆𝑜𝑖 𝜔𝜀 𝜀𝑍𝑖 𝑍𝑖 𝜀 4𝜋𝜀 2 ∑ 𝑐𝑖 𝑍𝑖2 2
= −( + ) ( ) ( ) … (10)
𝛼𝐹 𝛼𝐹 6𝐷𝐾𝑇 𝐾𝑖 6𝜋𝜂 300 𝐷𝐾𝑇

From equation (6) and (9), uniform velocity (or mobility) at infinite dilution is:

𝜀𝑍𝑖 𝜆𝑜𝑖 𝜀𝑍𝑖 300𝜆𝑜𝑖

= (or) =
300𝐾𝑖 𝛼𝐹 𝐾𝑖 𝛼𝐹

𝜀𝑍𝑖
Substituting the value of in equation (10), we get:
𝐾𝑖

1⁄
𝜆𝑖 𝜆𝑜𝑖 𝜔𝜀 300𝜆𝑜𝑖 𝑍𝑖 𝜀 4𝜋𝜀 2 ∑ 𝑐𝑖 𝑍𝑖2 2
= −( + ) ( ) ( ) … (11)
𝛼𝐹 𝛼𝐹 6𝐷𝐾𝑇 𝛼𝐹 6𝜋𝜂 300 𝐷𝐾𝑇

For complete dissociation in case of strong electrolyte, 𝛼 = 1.

1⁄
𝜆𝑖 𝜆𝑜𝑖 𝜔𝜀 300𝜆𝑜𝑖 𝑍𝑖 𝜀 4𝜋𝜀 2 ∑ 𝑐𝑖 𝑍𝑖2 2
= −( + ) ( ) ( )
𝐹 𝐹 6𝐷𝐾𝑇 𝐹 6𝜋𝜂 300 𝐷𝐾𝑇

1⁄ 1⁄
300𝜔𝜀𝜆𝑜𝑖 𝐹𝑍𝑖 𝜀 4𝜋𝜀 2 2 2
𝜆𝑖 = 𝜆𝑜𝑖 −( + ) ( ) ( ) (∑ 𝑐𝑖 𝑍𝑖2 )
6𝐷𝐾𝑇 6𝜋𝜂 300 𝐷𝐾𝑇

1⁄
𝑜 300𝜔𝜀𝜆𝑜𝑖 𝐹𝑍𝑖 𝜀 4𝜋𝜀 2 2
1⁄
𝜆𝑖 = 𝜆𝑖 − ( + ) ( ) ( ) (𝑐+ 𝑍+2 + 𝑐− 𝑍−2 ) 2 ⋯ (12)
6𝐷𝐾𝑇 6𝜋𝜂 300 𝐷𝐾𝑇

∵ For uni-univalent electrolyte, ∑ 𝑐𝑖 𝑍𝑖2 = 𝑐+ 𝑍+2 + 𝑐− 𝑍−2

Substituting the value of constants in equation (12), we get:

9.90 × 105 𝜔𝜆𝑜𝑖 29.15 𝑍𝑖 1⁄

𝜆𝑖 = 𝜆𝑜𝑖 − ( 3 + 1 )
(𝑐+ 𝑍+2 + 𝑐− 𝑍−2 ) 2
(𝐷𝑇) ⁄2 𝜂 (𝐷𝑇) ⁄2

9.90 × 105 𝜔 29.15 (𝑍+ + 𝑍− )

(or) 𝜆𝑖 = 𝜆𝑜𝑖 − ( 3⁄ × 𝜆𝑜𝑖 + 1 ) √(𝑐+ 𝑍+ 𝑍+ + 𝑐− 𝑍− 𝑍− )
(𝐷𝑇) 2 𝜂 (𝐷𝑇) ⁄2

9.90 × 105 𝜔 29.15 (𝑍+ + 𝑍− )

∴ 𝜆𝑖 = 𝜆𝑜𝑖 −( 3⁄ × 𝜆𝑜𝑖 + 1⁄ ) √𝑐(𝑍+ + 𝑍− )
(𝐷𝑇) 2 𝜂 (𝐷𝑇) 2

∵ 𝑐 (𝑖𝑛 𝑔𝑚 − 𝑒𝑞/𝐿) = 𝑐𝑖 × 𝑍𝑖

12
For uni-univalent ion, 𝑍+ = 𝑍− = 1 𝑎𝑛𝑑 𝜔 = 2 − √2

8.2 × 105 𝑜 82.4

(or) 𝜆 = 𝜆𝑜 − ( 3⁄ × 𝜆𝑖 + 1 ) √𝑐 … (13)
(𝐷𝑇) 2 𝜂 (𝐷𝑇) ⁄2

(or) 𝜆 = 𝜆𝑜 − (𝐴 + 𝐵𝜆𝑜𝑖 ) √𝑐 … (14)

82.4 8.2 × 105

where, 𝐴 = 1⁄ and 𝐵 = 3⁄
𝜂 (𝐷𝑇) 2 (𝐷𝑇) 2

The value of A and B for water at 25°C was found to be 60.2 and 0.224 respectively.

∴ 𝜆 = 𝜆𝑜 − [60.2 + 0.224𝜆𝑜𝑖 ]√𝑐 … (15)

The above equation is the Debye Hückel Onsager equation for uni-univalent electrolyte
dissolved in a solvent (water) at 25°C.
Equivalent conductance

HCl
KCl

AgNO3
NaCl

Concentration

Fig. 1.4: Molar Conductance against √𝒄

If the equation is correct, then by plotting molar conductance against √𝑐 would give a straight
line of slope is equal to [60.2 + 0.224𝜆𝑜𝑖 ] and intercept is equal to 𝜆𝑜𝑖 (Fig. 1.4).

VARIATION OF CONDUCTANCE WITH APPLIED POTENTIAL

Wein Effect (Conductance under High Potential Gradient)

When the applied potential of 20,000 V/cm is applied, an ion will move at a speed of 1 m/s.
That is the central ion will travel several times a thickness of the effective ionic atmosphere in
the time of relaxation.
As a result, the moving central ion is practically free from the opposite charge of the ionic
atmosphere (as there is no time for ions to build up ionic atmosphere to any extent). Under

13
LIMITATION OF DEBYE- HUCKEL- ONSAGER -EQUATION

Since the plot of conductance vs square root of the concentration is linear with
negative slope and positive intercept, it seems quite straightforward to study the strong
electrolytes. However, it has been observed that the equation is followed only up low and
moderate concentrations.

Figure 17. The comparison of theoretical and experimental conductance as a function of concentration for
some symmetric electrolytes.

It can be clearly seen that the theory and experiment move apart as the concentration increases. This is
simplybecause some approximation used to derive are Debye-Huckel-Onsager equation are not valid.